My guess is that the cat won't affect the running, since it's after all of the measuring gubbins... the symptoms you describe are indicative of a shot lambda sensor. I think the eggy smell is normal from a cat.
Incidentally, here's one for the MOT testers out there... is the cat actually tested in the MOT these days, or just the emissions? i.e., if you have a cat fitted, as required, but it's shot, can you still get a reading good enough to pass an MOT? Or put another way, would a car pass an emissions test without a cat, provided it was running well?
There's always one - and it's usually me!